Mathematics for the interested outsider

Last time, while talking about homotopiesas morphisms I said that I didn’t want to get too deeply into the reparameterization thing because it could get too complicated. But since when would I, of all people, shy away from 2-categories? In case it wasn’t obvious then, it’s because we’re actually going to extend in the other direction.

Given any two topological spaces and , we now don’t just have a set of continuous maps , we have a whole category consisting of those maps and homotopies between them. And I say that composition isn’t just a function that takes two (composable) maps and gives another one, it’s actually a functor.

So let’s say that we have maps , maps , and homotopies and . From this we can build a homotopy . The procedure is obvious: for any and , we just define

That is, the time- frame of the composed homotopy is the composition of the time- frames of the original homotopies. It should be straightforward to verify that this composition is (strictly) associative, and that the identity map — along with its identity homotopy — acts as an (also strict) identity.

What we need to show is that this composition is actually functorial. That is, we add maps and , change and to and , and add homotopies and . Then we have to check that

That is, if we stack onto and onto , and then compose them as defined above, we get the same result as if we compose with and with , and then stack the one onto the other.

This is pretty straightforward from a bird’s-eye view, but let’s check it in detail. On the left we have

Meanwhile, on the right we have

And so we do indeed have a 2-category with topological spaces as objects, continuous maps as 1-morphisms, and continuous homotopies as 2-morphisms. Of course, if we’re in a differential topological context we get a 2-category with differentiable manifolds as objects, smooth maps as 1-morphisms, and smooth homotopies as 2-morphisms.

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.